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Chapter 02 · Part II · Waves
Revised on ____________________
Chapter 02 · Waves and Sound

A pattern that travels, and the medium that stays

2 sheets
2 diagrams
28 results

Formulas and conditions only — no derivations, no solved numbers. The diagrams are the reader's own.

I

Travelling waves

THE PATTERN MOVES, THE PARTICLES DO NOT
I.Wave  y = A sin(k x − ω t)
II.Wave speed  v = ωk = f λ set by the medium, not the source
III.Particle speed  vy = A ω cos(k x − ω t) maximum Aω — a different quantity
IV.On a string  v = √( Tμ )
V.In a gas  v = √( γ R TM ) ∝ √T; pressure-independent
VI.In a solid rod  v = √( Yρ )
VII.Energy carried  P = 12 μ ω² A² v
VIII.Reflection  denser end inverts, free end does not
Wave speed and particle speed are unrelated numbers: their ratio is A k, which can be anything. A question about kinetic energy is always about the particle speed.
II

Standing waves and pipes

BOUNDARIES CHOOSE THE FREQUENCIES
I.Standing wave  y = 2A sin k x cos ω t no energy transport
II.Node spacing  λ2 antinode to nearest node: λ/4
III.String, both ends fixed  fn = n v2L all harmonics
IV.Open pipe  fn = n v2L all harmonics; ends are antinodes
V.Closed pipe  fn = (2n − 1) v4L odd harmonics only
VI.End correction  0.6 r per open end resonance tube: v = 2f(L₂ − L₁)
open pipeall harmonicsfₙ = nv/2Lclosed pipeodd harmonics onlyfₙ = (2n−1)v/4L
III

Intensity, loudness and beats

DECIBELS DO NOT ADD
I.Intensity  I ∝ A² f² ∝ 1/r² from a point source
II.Sound level  β = 10 log10 ( II0 ) dB I₀ = 10−12 W m−2
III.Doubling the intensity  + 3 dB ten times: +10 dB
IV.Beats  fbeat = |f1 − f2|
V.Beat carrier  f1 + f22 the pitch you hear
VI.Constructive  Δ = n λ path difference
VII.Destructive  Δ = (n + 12) λ
VIII.Two coherent sources  Imax : Imin = (a1 + a2)² : (a1 − a2)²
xdisplacement antinodepressure nodes(x) displ.p(x) press.
IV

The Doppler effect

THE MEDIUM IS THE REFERENCE
I.General  f′ = f v ± vov ∓ vs signs: toward each other raises f
II.Source moving  the wavelength in the medium changes
III.Observer moving  the rate of arrival changes the two are not equivalent
IV.Wind of speed w  replace v by v ± w
V.Mach number  M = vsrcv
VI.Shock cone  sin θ = 1M measured from the flight axis
Sound, 0 °C
331 m s−1
Sound, 20 °C
343 m s−1
Per °C
+ 0.61 m s−1
Audible range
20 Hz – 20 kHz
Threshold
I0 = 10−12 W m−2
Air, γ
1.4
V

Where the marks go

FOUR ERRORS, EVERY YEAR
Where marks are lost in this chapter
1Using the wave speed as the particle speed. The medium's element peaks at Aω; the pattern travels at ω/k.
2Swapping displacement and pressure. An open end is a displacement antinode and a pressure node.
3Adding decibels. 60 dB and 60 dB give 63 dB — convert to intensity, add, convert back.
4Writing fbeat = (f1 − f2)/2. That is the modulation rate; the beats heard are |f1 − f2|.